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Fearless Symmetry by Robert Gross, Avner Ash

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THE GREEKS HAD A NAME FOR IT 165
Conjugacy Classes
FACT: Any group H splits up into subsets C with the
following two properties:
1. Each subset C is obtained in the following way: You take
any group element, say x. Then you take all of the elements of
the group, call them g’s, and form the group products
g x g
1
. Notice that x itself is one of these products, because
e x e
1
= x (where e, as usual, is the neutral element). The
subset that consists of all these g x g
1
s is one of the C’s.
For example, x is an element of the C that you get by starting
with x. It does not matter which x in C you start with. You get
the same bunch of elements, namely C!
2. Any two elements of C have the same character value
under every representation r. In symbols: For any matrix
representation of the group H, call it r,ifx and y are in the
same C, then χ
r
(x) = χ
r
(y).
These subsets C are called the “conjugacy classes of H.” The C’s
are often quite large. The larger they are, the more the group law of
H fails to be commutative. At any rate, in a commutative group,
each conjugacy class contains only one element. This is because
if the group law is commutative, then g x g
1
= g g
1
x =
e x = x.
The definition of the C’s is actually accomplished just by prop-
erty 1. There is only one way of splitting up H into C’s that possess
property 1. The proof is just an exercise using the axioms of the
group law. Then property 2 is not too difficult to prove if you
know some linear algebra. The really amazing fact—and it is a
fact about representation theory—is that if x is in C and y is in
C
, which is some other conjugacy class, then there is some matrix
representation r such that χ
r
(x) = χ
r
(y). We will give an example of
this fact a little later for the group
{1,2,3}
.
Even more amazing is the important role that the characters
play in the number theory of Galois groups. Before we get to
that, though, we have some more to explain about group represen-
tations and their characters.

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